System of polynomial equations with cyclic symmetry

Question

solve a system of equations with unknowns x1, x2, xn where n >=2

Answer 1

Applying Buchberger's algorithm we obtain that there are the obvious solutions $$ (x_1,\ldots ,x_n)=(-1,\ldots,-1),\; (x_1,\ldots ,x_n)=(2,\ldots,2), $$ plus additional solutions given by $x_2=f(x_1),\ldots ,x_n=f_n(x_1)$ with polynomials $f_i\in \mathbb{Q}[x]$ of "high" degree, where $x_1$ has to satisfy another polynomial relation, i.e., is the root of a polynomial in one variable. For example, with $n=3$ we have that $x_1$ is a root of $$ 7168t^{24} - 96768t^{23} + 568512t^{22} - 1790784t^{21} + 2674128t^{ 20} + 1157184t^{19} - 12720844t^{18} + 20363868t^{17} + 741015t^{16} - 47236372t^{ 15} + 56887740t^{14} + 18319680t^{13} - 101773600t^{12} + 62822784t^{11} + 67667328t^{10} - 106537984t^9 + 3831552t^8 + 76778496t^7 - 35587072t^6 - 28164096t^5 + 24330240t^4 + 4390912t^3 - 7667712t^2 + 1048576. $$